Combing a hairy ball

The hair is combed uphill: north on the sphere, upwards on the torus, with two hills on the way. The middle of each hill is a small ice-blue ring; drag it and the hill moves to the point under your pointer, on the side of the surface facing you. The gold points are the spots that won't lie flat, the zeros in the corner, each labelled with its index: walk a small loop round it and count how many turns the hair makes, negative if it turns the other way. Hilltops and valleys are +1, passes are −1. At the start the hills are apart: each has its own hilltop, a little uphill of its ring, with a pass between them; drag the two hills together and that pass runs into one of the hilltops and both vanish. The corner finds each spot and its index, and counts the sum again cell by cell over the whole surface: on the sphere it is always 2. Raise the wind round the axis and the hair is blown round it; the spots drift, and some meet in pairs and vanish. They can end up anywhere, but the sum stays 2, so the sphere always keeps a spot that won't lie flat. Switch to the torus and the sum is 0; take the wind past 1 and it outweighs the slope everywhere: not a single spot is left, and the hair on the torus is combed flat.